📐 Concept diagram

03-06 - Complex Numbers

Phase: 3 | Subject: 03-06 Prerequisites: 01-08-quadratic-equations.md (discriminant), 02-07-unit-circle-and-radians.md Next subject: 03-07-sequences-and-series.md

Learning Objectives

By the end of this subject, you will be able to:

1. Understand why we need complex numbers (i = √(-1))
2. Perform arithmetic with complex numbers (add, subtract, multiply, divide)
3. Represent complex numbers on the complex plane (Argand diagram)
4. Find modulus and argument of a complex number
5. Convert between rectangular and polar forms

Core Content

Why Complex Numbers?

Not all quadratic equations have real solutions.

Example: x² + 1 = 0 x² = -1 x = ±√(-1)

There is no REAL number whose square is -1. So we define the imaginary unit:

$i = √(-1), so i² = -1
$

Any number of the form a + bi (where a, b are real) is a complex number.

• a = real part
• b = imaginary part
• If b = 0: it's a real number
• If a = 0: it's a pure imaginary number (bi)

Arithmetic

Addition and Subtraction

Add/subtract real and imaginary parts separately.

(2 + 3i) + (1 - 2i) = (2 + 1) + (3i - 2i) = 3 + i

Multiplication

Use FOIL (distributive law) and remember i² = -1.

(2 + 3i)(1 - 2i) = 2 - 4i + 3i - 6i² = 2 - i + 6 = 8 - i

Division

Multiply numerator and denominator by the COMPLEX CONJUGATE of the denominator.

Complex conjugate of a + bi is a - bi.

Key conjugate properties: - z · z̄ = |z|² (product of a number and its conjugate gives squared modulus) - For z = a + bi: (a+bi)(a-bi) = a² + b² = |z|² - The sum z + z̄ = 2a (twice the real part) - The difference z - z̄ = 2bi (twice the imaginary part)

Division using conjugate:

(3 + 2i)/(1 - i) = (3 + 2i)(1 + i) / ((1 - i)(1 + i)) = (3 + 3i + 2i + 2i²) / (1 - i²) = (3 + 5i - 2) / (1 + 1) = (1 + 5i) / 2 = 1/2 + (5/2)i

Complex Plane (Argand Diagram)

• Horizontal axis: Real part (Re)
• Vertical axis: Imaginary part (Im)

Example: 3 + 4i is plotted at point (3, 4)

Modulus and Argument

For z = a + bi:

Modulus: |z| = √(a² + b²) (distance from origin) Argument: arg(z) = θ where tan(θ) = b/a (angle from positive real axis)

Example: z = 3 + 4i |z| = √(9 + 16) = 5 arg(z) = tan⁻¹(4/3) ≈ 53.13°

Polar Form

z = a + bi can be written as:

$z = r·(cos(θ) + i·sin(θ)) = r·cis(θ)
$

where r = |z| and θ = arg(z).

Example: z = 3 + 4i r = 5, θ = tan⁻¹(4/3) ≈ 53.13° z = 5·cis(53.13°)

Euler's Formula

$e^(iθ) = cos(θ) + i·sin(θ)
$

So z = r·e^(iθ)

This connects complex numbers to exponentials and is CRITICAL in advanced math and physics.

De Moivre's Theorem

For any integer n:

$(r·cis(θ))^n = r^n·cis(nθ)
$

Example: (cos(30°) + i·sin(30°))³ = cos(90°) + i·sin(90°) = 0 + i(1) = i

Finding roots: To find the n-th roots of z = r·cis(θ):

z^(1/n) = r^(1/n)·cis((θ + 2πk)/n) for k = 0, 1, ..., n-1

Key Terms

• 03 06 Complex Numbers
• Addition and Subtraction
• Arithmetic
• Complex Plane (Argand Diagram)
• Complex conjugate of a + bi is a - bi.
• Correct: A)
• Correct: B)
• De Moivre's Theorem
• Division
• Euler's Formula
• Example 1: Simplify (1 + i)²
• Example 2: Divide (2 + i)/(1 - 3i)

Worked Examples

Example 1: Simplify (1 + i)²

(1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i

Example 2: Divide (2 + i)/(1 - 3i)

Multiply by conjugate: (2 + i)(1 + 3i) / ((1 - 3i)(1 + 3i)) Numerator: 2 + 6i + i + 3i² = 2 + 7i - 3 = -1 + 7i Denominator: 1 + 9 = 10 Result: (-1 + 7i)/10 = -0.1 + 0.7i

Example 3: Modulus and argument

z = -1 + i√3 |z| = √(1 + 3) = 2 arg(z) = tan⁻¹(√3 / -1). In Quadrant II: θ = 180° - 60° = 120° = 2π/3

Example 4: Finding roots of a complex number

Find ALL cube roots of z = 8 (i.e., solve w³ = 8).

In polar form: 8 = 8·cis(0). So r = 8, θ = 0.

Using the root formula: w_k = 8^(1/3)·cis((0 + 2πk)/3) for k = 0, 1, 2.

• k=0: w₀ = 2·cis(0) = 2(1 + i·0) = 2
• k=1: w₁ = 2·cis(2π/3) = 2(-1/2 + i·√3/2) = -1 + i√3
• k=2: w₂ = 2·cis(4π/3) = 2(-1/2 - i·√3/2) = -1 - i√3

The three cube roots of 8 are evenly spaced on a circle of radius 2.

Quiz

Q1: What does the concept of Addition and Subtraction primarily refer to in this subject?

A) The definition and application of Addition and Subtraction B) A computational error related to Addition and Subtraction C) A visual representation of Addition and Subtraction D) A historical anecdote about Addition and Subtraction

Correct: A)

• If you chose A: Addition and Subtraction is defined as: the definition and application of addition and subtraction. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Addition and Subtraction is defined as: the definition and application of addition and subtraction. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Addition and Subtraction is defined as: the definition and application of addition and subtraction. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Addition and Subtraction is defined as: the definition and application of addition and subtraction. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Arithmetic?

A) It is used to arithmetic in mathematical analysis B) It is primarily a historical notation system C) It is used only in advanced research contexts D) It replaces all other methods in this domain

Correct: A)

• If you chose A: Arithmetic serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose B: This is incorrect. Arithmetic serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Arithmetic serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Arithmetic serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Complex Plane (Argand Diagram) is TRUE?

A) Complex Plane (Argand Diagram) is not related to this subject B) Complex Plane (Argand Diagram) is mentioned only as a historical footnote C) Complex Plane (Argand Diagram) is an advanced topic beyond this subject's scope D) Complex Plane (Argand Diagram) is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. Complex Plane (Argand Diagram) is a fundamental concept covered in this subject. This subject covers Complex Plane (Argand Diagram) as part of its core content.
• If you chose B: This is incorrect. Complex Plane (Argand Diagram) is a fundamental concept covered in this subject. This subject covers Complex Plane (Argand Diagram) as part of its core content.
• If you chose C: This is incorrect. Complex Plane (Argand Diagram) is a fundamental concept covered in this subject. This subject covers Complex Plane (Argand Diagram) as part of its core content.
• If you chose D: Complex Plane (Argand Diagram) is a fundamental concept covered in this subject. This subject covers Complex Plane (Argand Diagram) as part of its core content. Correct!

Q4: Based on the worked examples in this subject, what is the correct result?

A) -0.1 + 0.7i B) A different result from a common mistake C) An unrelated numerical value D) The inverse of the correct answer

Correct: A)

• If you chose A: The worked examples show that the result is -0.1 + 0.7i. The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is -0.1 + 0.7i. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is -0.1 + 0.7i. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is -0.1 + 0.7i. The other options represent common errors.

Q5: How are Complex Plane (Argand Diagram) and Division related?

A) Complex Plane (Argand Diagram) is the inverse of Division B) Complex Plane (Argand Diagram) is a special case of Division C) Complex Plane (Argand Diagram) and Division are closely related concepts D) Complex Plane (Argand Diagram) and Division are completely unrelated topics

Correct: C)

• If you chose A: This is incorrect. Both Complex Plane (Argand Diagram) and Division are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Complex Plane (Argand Diagram) and Division are covered in this subject as interconnected topics.
• If you chose C: Both Complex Plane (Argand Diagram) and Division are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Complex Plane (Argand Diagram) and Division are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Why Complex Numbers??

A) A common mistake is confusing Why Complex Numbers? with a similar concept B) The main error with Why Complex Numbers? is using it when it is not needed C) Why Complex Numbers? is always computed the same way in all contexts D) Why Complex Numbers? has no common misconceptions

Correct: A)

• If you chose A: Students often confuse Why Complex Numbers? with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Why Complex Numbers? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Why Complex Numbers? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Why Complex Numbers? with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Multiplication?

A) Multiplication is not practically useful B) Apply Multiplication to solve problems in this subject's domain C) Avoid Multiplication unless explicitly instructed D) Use Multiplication only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. Multiplication is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Multiplication is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Multiplication is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Multiplication is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. i³ = ? Answer: i³ = i²·i = -1·i = -i

2. (2 + 3i) + (4 - i) Answer: 6 + 2i

3. (1 + i)(1 - i) Answer: 1 - i² = 1 + 1 = 2

4. |3 - 4i| Answer: √(9 + 16) = 5

5. Convert 1 + i to polar form Answer: r = √2, θ = π/4. z = √2·cis(π/4)

Summary

Key takeaways:

• i = √(-1), i² = -1, i³ = -i, i⁴ = 1
• Complex number: z = a + bi
• Conjugate of a + bi is a - bi (use for division)
• Complex plane: (a, b) corresponds to a + bi
• Modulus |z| = √(a² + b²)
• Argument arg(z) = angle from positive real axis
• Polar form: z = r·cis(θ)
• Euler's formula: e^(iθ) = cos(θ) + i·sin(θ)
• De Moivre: (r·cis(θ))^n = r^n·cis(nθ)

Pitfalls

• Forgetting i² = -1 during multiplication: When expanding complex products, students often treat i as a regular variable and leave i² unreplaced. Remember that i² = -1, i³ = -i, and i⁴ = 1. The simplification step that converts i² to -1 is critical for obtaining the correct real and imaginary parts.
• Using the wrong conjugate in division: The conjugate of a + bi is a - bi. When dividing, you multiply numerator AND denominator by the conjugate of the DENOMINATOR. A common mistake is using the conjugate of the numerator instead, or forgetting to multiply the numerator as well.
• Ignoring the quadrant when finding the argument: arg(z) = tan⁻¹(b/a) gives the reference angle, but you must adjust for the quadrant of (a, b) on the complex plane. For z = -1 + i, tan⁻¹(1/(-1)) ≈ -45°, but the correct argument is 135° (Quadrant II). Always sketch the point to confirm.
• Confusing arg(z) and Arg(z): The principal argument Arg(z) is restricted to (-π, π] or [0, 2π), while arg(z) represents all possible values differing by multiples of 2π. When using De Moivre or finding roots, ensure you're using the appropriate form for the context.
• Assuming (a + bi)^n = a^n + ib^n: This is NOT true. Powers of complex numbers follow De Moivre's theorem in polar form or binomial expansion with careful handling of i² = -1. For example, (1 + i)² = 2i, not 1 + i² = 0.

Next Steps

Next up: 03-07-sequences-and-series.md